Quadrangulations: Constructions and Geodesics

Wednesday, Jan 29, 2020

We previously worked with general two-complexes, including those representing surfaces. We now specialize to the first of the three special representations that are most useful to us - non-positive quadrangulations. Here we discuss what these are, how to construct these and the geodesics for these.

Non-positive quadrangulations

For a closed surface $\Sigma$, a non-positive quadrangulation is a two-complex homeomorphic to $\Sigma$ such that

1. Every face has four sides.
2. The degree of (i.e., number of edges terminating in) each vertex is at least $5$.

Constructing quadrangulations

Start with a $2$-complex $\Delta$ representing $\Sigma$. We describe a general procedure for constructing quadrangulations $Q$. If the $2$-complex $\Delta$ has one vertex and one face, and the surface has genus at least $2$, then the quadrangulation $Q$ we construct is automatically negatively curved.

The quadrangulation is the two complex $Q$ given in terms of $\Delta$ as follows.

1. Vertices: the vertices of $Q$ are the vertices of $\Delta$ together with a vertex $b_f$ for each face $f$ of $\Delta$ (thought of as the barycenter of $f$).
2. Edges: For a face $f$ with $n$ sides, with boundary $e_0(f), e_1(f), \dots, e_{n-1}(f)$, we have $n$ edges $\theta_i(f)$, $0 \leq i < n$, with $\theta_i$ starting at $b_f$ and ending at $e_i(f)$. Note that the edges of $\Delta$ are not edges of $Q$.
3. Faces: There is one face $\Phi(\eta)$ in $Q$ for each edge-pair $(\eta, \bar\eta)$, where bar denotes flip. Namely, we have unique faces $f$ and $g$ and indices $i$ and $j$ so that $\eta=e_i(f)$ and $\bar\eta = e_j(g)$. The corresponding face $\Phi(\eta)$ is the quadrilateral with boundary $\theta_i(f)$, $\bar\theta_{j + 1}(g)$, $\theta_j (g)$, $\bar\theta_i(f)$ (check this before implementing).

Geodesics

Fix a non-positive quadrangulation, so each vertex has degree at least $5$, and let $\eta$ be an edge. The left edge $\theta$ following $\eta$ is the successor in the unique face containing $\eta$ as a directed edge. This is the next edge in the sharpest left turn. We call going from $\eta$ to $\theta$ a left turn.

The slight left turn is the next sharpest left turn, obtained by taking a left turn, flipping and taking a left turn again (remember that this is a 5 or more pointed junction, so the terminology is justified). We can similarly define right turn and slight right turn (using flips and predecessors) as indicated in the previous note.

If an edge path has a segment where we make a left turn, followed by $0$ or more slight left turns and then another left turn, it is not a geodesic, and similarly with rights in place of lefts. Indeed in both these cases there is a clear shortening transformation that keeps the ends fixed and is a homotopy. We can apply such transformations till there are no obvious ones left.

The key point about non-positive curvature is that if there are no shortenings of the above form, the curve we have is a geodesic. Further geodesics are almost unique, and there is indeed a canonical leftmost geodesics. Many questions can be addressed using these.